Modules with serial Noetherian endomorphism rings
type: Item
instance of: scholarly article
title: Modules with serial Noetherian endomorphism rings
zbMATH Open document ID: 0693.16019
DOI: 10.1016/0021-8693(89)90264-0
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: A module is called serial if its submodules are linearly ordered with respect to inclusion. A ring \(\Lambda\) is said to be right (left) serial if \(\Lambda_{\Lambda}\) \((_{\Lambda}\Lambda)\) is a direct sum of serial submodules, and \(\Lambda\) is serial if it is both left and right serial. The paper delivers a characterization of generators with serial Noetherian endomorphism rings. Due to Warfield's structure theorem for serial Noetherian rings it suffices to obtain such a characterization separately for serial Artinian and serial prime Noetherian rings. Theorem 1. Let \(\Lambda\) be a serial Artinian ring and \(M=\oplus M_ i\) a \(\Lambda\)-module with indecomposable \(M_ i\). Then End M is serial Artinian if no \(M_ i\) is isomorphic to a proper subfactor of any \(M_ j\). Theorem 2. Let \(G_{\Lambda}\) be a generator. Then, the endomorphism ring of G is serial prime Noetherian if \(\Lambda\) is serial prime Noetherian and G is finitely generated projective.
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16P20
zbMATH DE Number: 4134259
zbMATH Keywords: direct sum of serial submodules
zbMATH Keywords: generators
zbMATH Keywords: serial Noetherian endomorphism rings
zbMATH Keywords: serial prime Noetherian rings
zbMATH Keywords: serial Artinian ring
author: Kunio Yamagata
reviewed by: I. Kh. Bekker
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90264-0
OpenAlex ID: W2051717508
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description: scientific article; zbMATH DE number 4134259